In this talk we present a survey of our research in the past decade on error bounds and convergence rates for deterministic as well as stochastic fixed point iterations, including the Krasnoselskii-Mann, Halpern, and similar iterative methods. 

We will emphasize the essential role played by techniques of optimal transport and Markov chains with rewards in obtaining tight error bounds. We will also discuss a few examples that illustrate how these error bounds are used to analyze the computational complexity of known algorithms in optimization and reinforcement learning for Markov decision processes, and how they can lead to design new algorithms with better complexity guarantees.

This talk outlines a novel class of high-order methods for the efficient numerical evaluation of volume potentials (VPs) defined by volume integrals over complex geometries. Inspired by the Density Interpolation Method (DIM) for boundary integral operators, the proposed methodology leverages Green’s third identity and a local polynomial interpolation of the density function to recast a given VP as a linear combination of surface-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated inside and outside the integration domain using existing methods (e.g. DIM), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules to integrate over structured or unstructured domain decompositions without local numerical treatment at and around the kernel singularity. The proposed methodology is flexible, easy to implement, and fully compatible with well-established fast algorithms such as the Fast Multipole Method and H-matrices, enabling VP evaluations to achieve linearithmic computational complexity. To demonstrate the merits of the proposed methodology, we applied it to the Nyström discretization of the Lippmann-Schwinger volume integral equation for frequency-domain Helmholtz scattering problems in piecewise-smooth variable media.

 

This is joint work with Thomas G. Anderson (Rice University), Luiz Faria (ENSTA Paris), and Marc Bonnet (ENSTA Paris).

 

Optimization problems are often solved using constraint programming or mixed-integer programming techniques, using enumeration or branch-and-bound techniques. It is well known that if the problem formulation is very symmetric, there may exist many symmetrically equivalent solutions to the problem. Without handling the symmetries, traditional solving methods have to check many symmetric parts of the solution space, which comes at a high computational cost. Handling symmetries in optimization problems is thus essential for devising efficient solution methods.

The presentation will introduce and review common methods for solving mathematical programs, the concepts of symmetries in mathematical programs, and how symmetries can be handled. Then, the main results of Jasper's dissertation are presented, along with computational results. In particular, the presentation focuses on a general framework for symmetry handling that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other in the literature, the framework allows to apply different symmetry handling methods simultaneously and thus outperform their individual effects. Moreover, most existing symmetry handling methods only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art methods implemented in the solver SCIP.

The Helmholtz equation for harmonic wave propagation is a widely used model for many acoustic phenomena, such as room acoustics, sonar, and biomedical ultrasound. The boundary element method (BEM) is one of the most efficient numerical methods to solve Helmholtz transmission problems and is based on boundary integral formulations that rewrite the volumetric partial differential equations into a representation of the acoustic fields in terms of surface potentials at the material interfaces. OptimUS (https://github.com/optimuslib/optimus) is a Python library centred around the BEM, which offers a user-friendly interface via Jupyter Notebooks, enabling the prediction of acoustic waves in piecewise homogeneous media in the frequency domain. 

 

This talk will provide an overview of the OptimUS interface, with a focus on case studies where objects are large relative to the wavelengths involved. This will include biomedical ultrasound, which has a growing number of therapeutic applications such as the treatment of cancers of the liver and kidney. The modelling of transcranial ultrasound neurostimulation, an emerging modality which may one day treat mental health conditions such as depression, will also be reviewed. Acoustic wave propagation into the uterus at audio range frequencies will be presented to provide awareness of the impact of everyday noise exposure on the developing fetus.

 

When studying complex systems that evolve over time, it is often not enough to just predict the future. We also want to know if equilibrium states are stable, how the system will behave on average over a long time, and how uncertainty impacts the dynamics. Techniques from polynomial optimization can be used to make these predictions when we have explicit models for the dynamics based on polynomial equations. But what if the model isn’t based on polynomial equations or, worse, we don't know the model at all?

 

In this talk, I will discuss how we can analyze dynamical systems using only data from measurements, with no need for an explicit mathematical model. The main idea is to combine polynomial optimization with a data analysis technique called extended dynamic mode decomposition, which is related to the celebrated Koopman operator. I will introduce these tools, show how they work together, and illustrate their effectiveness with examples related to stability and long-term behavior.

Fixed-point iterations provide a systematic method to approximate solutions for a wide range of problems, including operator equations, nonlinear equations, and optimization tasks. Recently, stochastic variants of these iterations have garnered attention due to their applicability in scenarios where the underlying operators are uncertain, noisy, or random. However, despite this growing interest, stochastic fixed-point iterations in non-Euclidean settings remain underexplored compared to related frameworks, such as stochastic monotone inclusions, variational inequalities, or stochastic optimization, which are predominantly studied in Euclidean spaces.

 

In this talk, we analyze the oracle complexity of stochastic Mann-type iterations designed to approximate fixed points of nonexpansive and contractive maps in a finite-dimensional space equipped with a general norm. We establish both upper and lower bounds on the convergence rate of the fixed-point residual and provide conditions for almost sure convergence of the iterates. Our findings have potential applications in reinforcement learning and bilevel optimization.

Prophet inequalities are a cornerstone in optimal stopping and online decision-making. Traditionally, they involve the sequential observation of n non-negative independent random variables and face irrevocable accept-or-reject choices. The goal is to provide policies that provide a good approximation ratio against the optimal offline solution that can access all the values upfront -- the so-called prophet value. In the prophet inequality over time problem, the decision-maker can commit to an accepted value for T units of time, during which no new values can be accepted. This creates a trade-off between the duration of commitment and the opportunity to capture potentially higher future values.

 

In this work we provide optimal approximation guarantees for the IID setting. We show a single-threshold algorithm that achieves an approximation ratio of (1+1/e^2)/2≈0.567, and we prove that no single-threshold algorithm can surpass this guarantee. Then, for each n, we prove it is possible to compute the tight worst-case approximation ratio of the optimal dynamic programming policy for instances with n values by solving a convex optimization program. A limit analysis of the first-order optimality conditions yields a nonlinear differential equation showing that the optimal dynamic programming policy's asymptotic worst-case approximation ratio is ≈0.618. Joint work with Sebastian Pérez-Salazar (Rice University, USA).

The advent of generative AI has turbocharged the development of a myriad of commercial applications, and it has slowly started to permeate into scientific computing. In this talk we discussed how recasting the formulation of old and new problems within a probabilistic approach opens the door to leverage and tailor state-of-the-art generative AI tools. As such, we review recent advancements in Probabilistic SciML – including computational fluid dynamics, inverse problems, and particularly climate sciences, with an emphasis on statistical downscaling.

 

Statistical downscaling is a crucial tool for analyzing the regional effects of climate change under different climate models: it seeks to transform low-resolution data from a (potentially biased) coarse-grained numerical scheme (which is computationally inexpensive) into high-resolution data consistent with high-fidelity models.

 

We recast this problem in a two-stage probabilistic framework using unpaired data by combining two transformations: a debiasing step performed by an optimal transport map, followed by an upsampling step achieved through a probabilistic conditional diffusion model. Our approach characterizes conditional distribution without requiring paired data and faithfully recovers relevant physical statistics, even from biased samples.

 

We will show that our method generates statistically correct high-resolution outputs from low-resolution ones, for different chaotic systems, including well known climate models and weather data. We show that the framework is able to upsample up to 300x while accurately matching the statistics of relevant physical quantities – even when the low-frequency content of the inputs and outputs differs. This is a crucial yet challenging requirement that existing state-of-the-art methods usually struggle with.

A particularly prominent area of machine learning is generative modeling, in which one seeks to artificially generate new data from a given input. For instance, generating images from a chosen text prompt. Mathematically, this can be posed as finding a map that pushes a simple probability distribution, such as a standard Gaussian to a complicated probability distribution. New data can then be created by sampling random points from the simple distribution and then sending them through the constructed map.

 

Currently, one of the most popular paradigms used in generative modeling is diffusion modeling, where the map is created by observing the behavior of a diffusion equation applied to the given data. In this talk, I will give an introduction to diffusion modeling, discuss some of my work approaching the problem from a deterministic PDE perspective (as opposed to the more typical stochastic approach), and mention some interesting open questions in this area.

Terapias con estimulación eléctrica son usadas para tratar síntomas de diferentes desórdenes del sistema nervioso. En este contexto, el uso de señales de alta frecuencia ha recibido mucha atención debido a sus efectos en tejidos y células. En esta charla veremos cómo métodos matemáticos son útiles para abordar algunas preguntas relevantes cuando para la neurona se considera un modelo de FitzHugh-Nagumo. Acá la estimulación es a través de un término fuente en la ecuación diferencial ordinaria y el nivel de activación de la neurona está asociado con la existencia de potenciales de acción que son soluciones con un comportamiento específico. Una primera pregunta se relaciona con la efectividad de una técnica reciente llamada corrientes interferenciales, que combina dos señales de frecuencia kilohertz con el objetivo de lograr activación profunda. La segunda pregunta es sobre cómo evitar la activación inicial no deseada que se origina al comenzar a estimular con señales. Mostraremos resultados teóricos y computacionales usando métodos como promediado, análisis de Lyapunov, deformación casi-estática y otros.

We examine a class of mixed integer linear programming problems characterized by having a large set of precedence constraints and a small number of additional, “arbitrary” side constraints. These problems, which in a way are “almost” totally unimodular, are applicable in a wide array of scheduling tasks, including the well-known Resource-Constrained Project Scheduling Problem (RCPSP). RCPSPs and their variants are known to be extremely difficult to solve in practice. Moreover, they are of particular importance in the field of mining, where the scale of the problems can involve hundreds of millions of variables, posing a challenge for standard commercial solvers.

 

In this talk will describe these precedence-constrained optimization problems and discuss how understanding the optimal solution structure can inform the creation of specialized linear programming techniques that are more scalable than traditional algorithms. We will also describe new classes of cutting planes to strengthen linear relaxations. Applications of these methods will be showcased, ranging from scheduling for open pit and underground mines to adapting to uncertainties and integrating environmental objectives into scheduling practices.

 

This is joint work with Patricio Lamas, Eduardo Moreno and Orlando Rivera.

I will give a broad introduction to synthetic data and in particular their applications to Finance. I will focus on fair synthetic data based on creating a small coreset of the original dataset utilizing the Wasserstein distance while enforcing demographic parity. Experiments show the benefits of the approach and will also include reducing bias in LLMs. The talk will be broadly based on:

 

https://arxiv.org/abs/2401.00081

 

https://arxiv.org/abs/2311.05436

The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. In this talk, we will introduce the proximal Galerkin method and apply it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The proximal Galerkin framework is a natural consequence of the latent variable proximal point (LVPP) methodology, which is a stable and robust alternative to the interior point method that will also be introduced in this talk. LVPP can be viewed as a low-iteration complexity, infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout the talk, we will arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and an infinite-dimensional Lie group; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. This is joint work with T.M. Surowiec.